Express 1994 as a sum of consecutive positive integers and show that this is the only way to do it.

497+498+499+500

 

since 1994 has factors 1,2,997 and 1994

and the sum of this arithmetic series is (n/2)*(2a +n-1) then either n/2 or n is a factor.

 

ie n = 1,2, 997, 1994 or 2,4, 1994 or 3988

 

since the large values of n give negative a and n = 2 must give an odd number and n = 1 is the trivial case n = 4 is the only possibility